PHYSICAL REVIEW B VOLUME 45, NUMBER 13 Fluxon viscosity in high-T, superconductors 1 APRIL 1992-I Michael Golosovsky, Yehuda Naveh, and Dan Davidov Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel (Received 12 August 1991) We present a phenomenological model for the frequency-dependent fluxon complex resistivity in which the Quxon viscosity g is a central parameter. We use the high-frequency limit of this model to ex- tract g from microwave transmission measurements through thin superconducting films in the presence of a magnetic field. The measured value of g allows a critical check of the model at low frequencies. The viscosity is anisotropic and consistent with anisotropic efFective mass ratios of y =(I, /m, )' =5-8 for Y-Ba-Cu-0 and y =12— 15 for Bi-Sr-Ca-Cu-O. It is now accepted that the complex resistance of high- T, superconductors is associated with fluxon dynamics and strongly depends on magnetic field, frequency, and temperature. The very early model of Gittleman and Rosenblum' assumed oscillations of damped fluxons in a harmonic pinning potential. According to this model the fluxon resistivity is dissipative at high frequencies but it is nondissipative at low frequencies. However, this model' was developed for zero temperature and cannot account for the low-frequency dissipative properties, particularly for the thermally activated dc resistivity which is very pronounced in high-T, superconductors. The dc resis- tivity of high-T, superconductors in the presence of a magnetic field has been successfully explained by several models which assume flux hopping between adjacent pinning sites. These models take into account pinning forces but do not depend explicitly on the fluxon viscosi- ty. Clearly, there is a need for a more general model for the fluxon dissipation at different frequencies and temper- atures. Initial attempts in this direction were made by Inui, Littlewood, and Coppersmith and by Martinoli et al. , who have used the analogy between a pinned fluxon and a Brownian particle in a potential well to cal- culate the low-frequency complex resistivity. The present paper extends the theoretical approach of Martinoli et a/. for wide frequency ranges and for different ternper- atures. The phenomenological model assumes uncor related motion of fluxons in a harmonic pinning potential U(x)= Up[1 — cos(qx)]. Here q ' is the pinning length and Uo is the amplitude of the pinning potential. Both of these parameters depend on temperature. Correlations in the fluxon motion may be accounted for by introducing magnetic-field-dependent Uo and q . A single pinning energy is assumed. The fluxons are regarded as "rigid rods" with length L and negligible mass. The driving force for fluxon motion is the Lorentz force F, „, =( J@GL /c) sing, where J is the current density, qr is the angle between the microwave current and the mag- netic field, and 40 is the flux quantum. The model also assumes the presence of a stochastic force F, & due to thermal fluctuations which satisfies (F, I, (t)F,b(0) ) =2gL T k(t5), where 5(t) is a 5 function, g is the temperature-dependent viscosity coe%cient per unit length, and k is the Boltzmann factor. The fluxon con- centration is assumed to be uniform. Therefore, the fluxon motion is two dimensional and can be described by the Langevin equation of motion: riLx+ Upq sin(qx) =F, „, +F, b, where x is the fluxon displacement. The moving fluxons produce an electric field E =xH/c, where x can be calcu- lated using Eq. (1). For an oscillating external current J = Jp exp(i tdt ), the rf resistivity associated with the fluxon motion is given by Z&(td) =E/J. In the limit of small current ( Jp (( Upqc /4GL) Eq. (1) can be solved ap- proximately by a continued-fraction expansion. One ob- tains, then, for the fluxon complex resistivity ZI(to) =RI 1+ IG(s) — 1 1 — i (td/cop) [Ip (s) — 1]IG(s)/I, (s) (cd/co, ) +Ip (s) pf Rf 1+(to/cd, ) Cd /Cd ~ Xf = RI [1 — Ip (s) ] 1+(td/td, ) where (3a) (3b) Ip(s)I & (s) CO) =COO Ip(s) — 1 (4) At low frequencies the pinning forces dominate (pinning regime) while at high frequencies the viscous forces dom- inate (flux-flow regime). The model predicts a crossover frequency [Eq. (4)] from the pinning regime to the fiux- flow regime. Far below T, the crossover frequency can be expressed as (2) where R&=H~ sintP~@p/71c, s = Up/kT, cop= Upq /riL, Ip(s) and It(s) are modified Bessel functions. ' Here q& is the angle between the magnetic field H and the mi- crowave current I „(see Fig. 1). The real part pI and the imaginary part XI of the fluxon complex resistivity, as calculated using Eq. (2), are given by the following ex- pressions: 45 7495 1992 The American Physical Society